L’objectif de ce TP est de traiter le cas d’un mélange gaussien (à deux composantes pour le moment).

Simulation d’un mélange

On peut commencer par simuler un mélange pour travailler avec des données “faciles” :

rmix = function(N,p,moy1,moy2,t1,t2){
  b = rbinom(N,1,p)
  x = (1-b)*rnorm(N,moy1,1/sqrt(t1))+(b)*rnorm(N,moy2,1/sqrt(t2))
  return(x)
  }
hist(rmix(1000,0.6,0,5,1,1),100,main = "Simulated Mixture")

Modèle à variables cachés

On travaille maintenant avec le modèle suivant :

Xi|Zi,m1,m2,t1,t2i.i.dN(mZi,1tZi)Zi|pi.i.d1+B(p)pBeta(a0,b0)miN(μ0,σ02),i=1,2τiGamma(k0,λ0),i=1,2

On peut initialiser les données :

N = 1000
true = list(p = 0.6,m1 = 0,m2 = 4,tau1 =1,tau2 = 5)##pour la simulation
x = rmix(1000,true$p,true$m1,true$m2,true$tau1,true$tau2)
h = list(a =1,b = 1, mu = 0,sig = 1000,k = 1,lam = 0.001)
l = list(p  = 1/2,m = c(0,0),tau = c(1,1),z = 1+rbinom(length(x),1,1/2))
init = function(x){
  h = list(a =1,b = 1, mu = 0,sig = 1000,k = 1,lam = 0.001)
  l = list(p  = 1/2,m = c(0,0),tau = c(1,1),z = 1+rbinom(length(x),1,1/2))
  return(list(h = h,l = l))
}
logvraiss= function(l,x){
  return( sum( log( dnorm(x,l$m[l$z],1/sqrt(l$tau[l$z])    )   )       )         ) 
}
logvraiss(l,x)
[1] -5851.625
logprior = function(l,h){
  val = sum(log(dbinom(l$z-1,1,l$p )))+
    log(dbeta(l$p,h$a,h$b))+
    sum(log(dnorm(l$m,h$mu,h$sig)  ))+
    sum(log(dgamma(l$tau,h$k,h$lam)))
  return(val)
}
logprior(l,h)
[1] -722.6181
logpost = function(l,x,h){
    return(logvraiss(l,x)+logprior(l,h))
}
logpost(l,x,h)
[1] -6574.243
step_MH = function(l,x,h,sigprop = c(0.01,0.01,0.01),nrunz = 50){
  ##variance prop codées en dur pour le moment
  sig_prop_m = sigprop[1]
  inv_prop_tau = 1/sigprop[2]
  inv_prop_p = 1/sigprop[3]
  lnew = l
  N = length(x)
  # print(logpost(lnew,x,h))
  ##ici, c'est bien couteux de recalculer tout à chaque fois -> on devrait calculer que pour x[i] la vraisemblance !!!!
  ##proposition m
  i = sample(2,1)
  lnew$m[i] = lnew$m[i]+rnorm(1,0,sig_prop_m)
  if (log(runif(1))>(logpost(lnew,x,h)-logpost(l,x,h)) ){
    ##on rejette avec proba 1-alpha,  ici la proposition est symétrique
    lnew$m[i] = l$m[i]
  }
  # print(logpost(lnew,x,h))
  if (lnew$m[2]<lnew$m[1]){
    ##si les m se sont croisés, on échange tout : m, tau, p et z
    lnew$m = rev(lnew$m)
    lnew$tau = rev(lnew$tau)
    lnew$p = 1-lnew$p
    lnew$z = 3-lnew$z
  }
  # print(logpost(lnew,x,h))
  ##proposition tau
  i = sample(2,1)
  lnew$tau[i] = rgamma(1,inv_prop_tau*l$tau[i]**2,inv_prop_tau *l$tau[i])
  log_ratio_prop = log(dgamma(l$tau[i]     ,inv_prop_tau*lnew$tau[i]**2,inv_prop_tau *lnew$tau[i]))-log(dgamma(lnew$tau[i]     ,inv_prop_tau*l$tau[i]**2,inv_prop_tau *l$tau[i]))
  if (log(runif(1))>(log_ratio_prop+logpost(lnew,x,h)-logpost(l,x,h)) ){
    ##on rejette avec proba 1-alpha. ici on prend en compte le ratio
    lnew$tau[i] = l$tau[i]
  }
  ##proposition z
  # for (i in 1:N){
  # nrunz = 50
  samp = sample(N,nrunz)
  for (i in samp){
    # if (runif(1)<1/2){
      # i = sample(N,1)
      lnew$z[i] = 3-l$z[i]
      if (log(runif(1))>(logpost(lnew,x,h)-logpost(l,x,h)) ){
        ##on rejette avec proba 1-alpha. ici la proposition est symétrique
        lnew$z[i] = l$z[i]
      } else{
        l$z[i] = lnew$z[i]
      }
    # }
  }
  # print(logpost(lnew,x,h))
  ##proposition p
  lnew$p = rbeta(1,inv_prop_p*l$p,inv_prop_p*(1-l$p))
  log_ratio_prop = log(dbeta(l$p,inv_prop_p*lnew$p,inv_prop_p*(1-lnew$p)))-log(dbeta(lnew$p,inv_prop_p*l$p,inv_prop_p*(1-l$p)))
    if (log(runif(1))>(log_ratio_prop+logpost(lnew,x,h)-logpost(l,x,h)) ){
    ##on rejette avec proba 1-alpha. ici on prend en compte le ratio
    lnew$p = l$p
    }
  return(lnew)
}
lnew = step_MH(l,x,h)
run_MH = function(nMH,x,...){
  linit = init(x)
  l = linit$l
  h = linit$h
  traj = matrix(0,nMH,5)
  colnames(traj) = c("p","m0","m1","tau0","tau1")
  nb = rep(0,length(x))
  for (i in 1:nMH){
    nb = nb+(l$z-1)
    l = step_MH(l,x,h,...)
    traj[i,] = c(l$p,l$m,l$tau)
  }
  class = nb/nMH
  return(list(traj = traj ,class =class,l = l))
}
nMH = 100
ltraj  = run_MH(nMH,x)
plot(1:nMH,ltraj$traj[,2],"l")
lines(1:nMH,ltraj$traj[,3],col = "red")

Pour un plot plus joli, on reprend la fonction de la dernière fois

plot_run = function(x,mem,l,i = NULL,nbpoints = 10000){
    if (is.null(i)){i = dim(mem)[1]}
    par(mfrow = c(1,2))
    hist(x,50,freq= F,main = "Current estimated density")
    curve((1-l$p)*dnorm(x,l$m1,1/sqrt(l$tau1))+(l$p)*dnorm(x,l$m2,1/sqrt(l$tau2)),add = T,col = "red",lty = 2,lwd=  2)
    curve((1-l$p)*dnorm(x,l$m1,1/sqrt(l$tau1)),add = T,col = "blue",lty = 3,lwd = 2)
    curve((l$p)*dnorm(x,l$m2,1/sqrt(l$tau2)),add = T,col = "green",lty = 3,lwd = 2)
    z1 = sample(i,nbpoints,replace = T)
    z2 = sample(i,nbpoints,replace = T)
    plot(z1,mem[z1,2]+2*1/sqrt(mem[z1,4])*runif(nbpoints,-1,1),ylim = range(x),col = "cyan",cex = 0.1,main = "MH trajectory with 2-sigma bands",ylab = "Trajectories for m1,m2",xlab= "Time")
    points(z2,mem[z2,3]+2*1/sqrt(mem[z2,5])*runif(nbpoints,-1,1),ylim = range(x),col = "yellow",cex = 0.1)
    lines(1:i,mem[1:i,2],ylim = range(x),col = "blue")
    lines(1:i,mem[1:i,3],col = "green")
}
init = function(x){
  h = list(a =1,b = 1, mu = 0,sig = 1000,k = 1,lam = 0.001)
  l = list(p  = 1/2,m = c(0,0),tau = c(1,1),z = 1+rbinom(length(x),1,1/2))
  # l = list(p  = 1/2,m = c(-1,1),tau = c(1,1),z = 1+(x>mean(x)))
  return(list(h = h,l = l))
}
N = 1000
true = list(p = 0.6,m1 = 0,m2 = 2,tau1 =1,tau2 = 5)##pour la simulation
x = rmix(1000,true$p,true$m1,true$m2,true$tau1,true$tau2)
nMH = 2000
sigprop =  c(0.5,0.1,0.01)
nrunz = 50
ltraj  = run_MH(nMH,x,sigprop,nrunz)
l = ltraj$l
lend = list(p = l$p,m1  = l$m[1],m2 = l$m[2],tau1 = l$tau[1],tau2 = l$tau[2])
plot_run(x,ltraj$traj,lend)

On peut alors voir les paramètres finaux :

cat("estimés : ",unlist(lend),"\n")
estimés :  0.5321751 0.4869726 1.986869 0.7883785 5.634391 
cat("théorique :",unlist(true),"\n")
théorique : 0.6 0 2 1 5 

On peut aussi observer la propotion du temps que chaque observation a passé avec Zi=1 :

s = sort(x,index.return = T)
plot(s$x,ltraj$class[s$ix],"l",main  = "Classification probability",xlab = "x",ylab = "Estimated P(Z(x)= 1)")

Et si on veut faire un petit film, il n’y a qu’à modifier les fonctions

library(animation)
plot_run = function(x,mem,class,l,i = NULL,nbpoints = 10000,MHtot = NULL){
    if (is.null(i)){i = dim(mem)[1]}
    if (is.null(MHtot)){MHtot = dim(mem)[1]}
    # par(mfrow = c(1,3))
    hist(x,50,freq= F,main = "Current estimated density")
    curve((1-l$p)*dnorm(x,l$m1,1/sqrt(l$tau1))+(l$p)*dnorm(x,l$m2,1/sqrt(l$tau2)),add = T,col = "red",lty = 2,lwd=  3)
    curve((1-l$p)*dnorm(x,l$m1,1/sqrt(l$tau1)),add = T,col = "blue",lty = 3,lwd = 3)
    curve((l$p)*dnorm(x,l$m2,1/sqrt(l$tau2)),add = T,col = "green",lty = 3,lwd = 3)
    z1 = sample(i,nbpoints,replace = T)
    z2 = sample(i,nbpoints,replace = T)
    plot(z1,mem[z1,2]+2*1/sqrt(mem[z1,4])*runif(nbpoints,-1,1),ylim = range(x),xlim = c(1,MHtot),col = "cyan",cex = 0.1,main = "MH trajectory with 2-sigma bands")
    points(z2,mem[z2,3]+2*1/sqrt(mem[z2,5])*runif(nbpoints,-1,1),ylim = range(x),col = "yellow",cex = 0.1)
    lines(1:i,mem[1:i,2],ylim = range(x),col = "blue")
    lines(1:i,mem[1:i,3],col = "green")
    s = sort(x,index.return = T)
    plot(s$x,class[s$ix]/i,"l")
}
run_MH = function(nMH,x,pas_anim,...){
  linit = init(x)
  l = linit$l
  h = linit$h
  traj = matrix(0,nMH,5)
  colnames(traj) = c("p","m0","m1","tau0","tau1")
  nb = rep(0,length(x))
  traj[1,] = c(l$p,l$m,l$tau)
  for (i in 2:nMH){
    nb = nb+(l$z-1)
    l = step_MH(l,x,h,...)
    lnow = list(p = l$p,m1  = l$m[1],m2 = l$m[2],tau1 = l$tau[1],tau2 = l$tau[2])
    if ((i%%pas_anim)==2){
      plot_run(x,traj[1:i,],class = nb,l = lnow,i=i,MHtot =nMH )
      ani.record()
    }
    traj[i,] = c(l$p,l$m,l$tau)
  }
  class = nb/nMH
  return(list(traj = traj ,class =class,l = l))
}
# 
# N = 100
# true = list(p = 0.6,m1 = 0,m2 = 2,tau1 =1,tau2 = 5)##pour la simulation
# x = rmix(1000,true$p,true$m1,true$m2,true$tau1,true$tau2)
# sigprop =  0.1*c(0.5,0.1,0.01)
# nrunz = 20
# nMH = 500
# pas_anim = 10
# par(mfrow = c(1,3))
# ani.options(interval = 0.05, nmax = floor(nMH/pas_anim))
# ani.record(reset = TRUE)
# ltraj  = run_MH(nMH,x,pas_anim= pas_anim,sigprop,nrunz)
# saveGIF(ani.replay(),movie.name = "animation_MH.gif")

Commentaires et extensions

On peut aussi observer la chose suivante :

π(m1|X,Z,τ,p)π(m1)i=1N(exp(τ1(xim1)22))11zi=1exp((m1μ0)22σ02i=1N11zi=1τ1(xim1)22)

Par propriété de conjugaison de loi, on voit alors que

m1|X,Z,τ,pN(μ0σ02+τ1i=1N11zi=1xi1σ02+τ1i=1N11zi=1,11σ02+τ1i=1N11zi=1)

Et de même pour tous les autres paramètres. Cela incite à reprendre tout notre code pour calculer seulement cela à chaque étape, sans devoir recalculer la vraisemblance.

Voilà un bon exercice pour votre projet !

Et avec les packages

setwd("/home/espinasse/Bureau/TAF/Seafile/Cours/stat bayesiennes/tp R 2020/old_site")
library(rjags)
Le chargement a nécessité le package : coda
Linked to JAGS 4.3.0
Loaded modules: basemod,bugs
N = 1000
true = list(p = 0.6,m1 = 0,m2 = 2,tau1 =1,tau2 = 5)##pour la simulation
u = rmix(10000,true$p,true$m1,true$m2,true$tau1,true$tau2)
m <- jags.model("model_mixture.bug", data=list(x = u, N=N))
Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 1000
   Unobserved stochastic nodes: 1005
   Total graph size: 14010

Initializing model
f = coda.samples(m,c("m","tau","p"),n.iter = 1000)

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traj = f[[1]]
summary(f)

Iterations = 1:1000
Thinning interval = 1 
Number of chains = 1 
Sample size per chain = 1000 

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

           Mean      SD  Naive SE Time-series SE
m[1]   -0.02481 0.14448 0.0045689       0.018585
m[2]    1.95753 0.05871 0.0018567       0.005077
p       0.62579 0.02915 0.0009219       0.003282
tau[1]  0.92491 0.13191 0.0041714       0.015436
tau[2]  5.39071 0.60625 0.0191714       0.064464

2. Quantiles for each variable:

          2.5%     25%      50%     75%  97.5%
m[1]   -0.2420 -0.1075 -0.03618 0.04833 0.1887
m[2]    1.9100  1.9480  1.96294 1.97752 2.0037
p       0.5721  0.6085  0.62655 0.64461 0.6777
tau[1]  0.6953  0.8372  0.91221 1.01074 1.2072
tau[2]  4.5220  5.1226  5.40535 5.71578 6.3382
par(mfrow = c(5,2))
# for (i in 1:5){
#   plot(1:nrow(traj),traj[,i],"l",main = colnames(traj)[i])
# }
plot(f)

---
title: "R Notebook"
output: html_notebook
---

L'objectif de ce TP est de traiter le cas d'un mélange gaussien (à deux composantes pour le moment). 


# Simulation d'un mélange
On peut commencer par simuler un mélange pour travailler avec des données "faciles" :

```{r}
rmix = function(N,p,moy1,moy2,t1,t2){
  b = rbinom(N,1,p)
  x = (1-b)*rnorm(N,moy1,1/sqrt(t1))+(b)*rnorm(N,moy2,1/sqrt(t2))
  return(x)
  }
hist(rmix(1000,0.6,0,5,1,1),100,main = "Simulated Mixture")
```


# Modèle à variables cachés 


On travaille maintenant avec le modèle suivant :

$$\begin{align*}
X_i|Z_i,m_1,m_2,t_1,t_2& \sim_{i.i.d} \mathcal{N}\Big(m_{Z_i},\frac{1}{t_{Z_i}}\Big)
\\Z_i|p &\sim_{i.i.d} 1+\mathcal{B}(p)
\\ p &\sim Beta(a_0,b_0)
\\ m_i &\sim \mathcal{N}\Big(\mu_0,\sigma_0^2\Big), i = 1, 2
\\ \tau_i &\sim Gamma(k_0,\lambda_0), i = 1,2
\end{align*}$$


On peut initialiser les données :
```{r}
N = 1000
true = list(p = 0.6,m1 = 0,m2 = 4,tau1 =1,tau2 = 5)##pour la simulation
x = rmix(1000,true$p,true$m1,true$m2,true$tau1,true$tau2)
h = list(a =1,b = 1, mu = 0,sig = 1000,k = 1,lam = 0.001)
l = list(p  = 1/2,m = c(0,0),tau = c(1,1),z = 1+rbinom(length(x),1,1/2))
```


```{r}

init = function(x){
  h = list(a =1,b = 1, mu = 0,sig = 1000,k = 1,lam = 0.001)
  l = list(p  = 1/2,m = c(0,0),tau = c(1,1),z = 1+rbinom(length(x),1,1/2))
  return(list(h = h,l = l))
}

logvraiss= function(l,x){
  return( sum( log( dnorm(x,l$m[l$z],1/sqrt(l$tau[l$z])    )   )       )         ) 
}

logvraiss(l,x)

logprior = function(l,h){
  val = sum(log(dbinom(l$z-1,1,l$p )))+
    log(dbeta(l$p,h$a,h$b))+
    sum(log(dnorm(l$m,h$mu,h$sig)  ))+
    sum(log(dgamma(l$tau,h$k,h$lam)))
  return(val)
}

logprior(l,h)

logpost = function(l,x,h){
    return(logvraiss(l,x)+logprior(l,h))
}

logpost(l,x,h)
```




```{r}
step_MH = function(l,x,h,sigprop = c(0.01,0.01,0.01),nrunz = 50){
  ##variance prop codées en dur pour le moment
  sig_prop_m = sigprop[1]
  inv_prop_tau = 1/sigprop[2]
  inv_prop_p = 1/sigprop[3]
  lnew = l
  N = length(x)
  # print(logpost(lnew,x,h))
  ##ici, c'est bien couteux de recalculer tout à chaque fois -> on devrait calculer que pour x[i] la vraisemblance !!!!
  ##proposition m
  i = sample(2,1)
  lnew$m[i] = lnew$m[i]+rnorm(1,0,sig_prop_m)
  if (log(runif(1))>(logpost(lnew,x,h)-logpost(l,x,h)) ){
    ##on rejette avec proba 1-alpha,  ici la proposition est symétrique
    lnew$m[i] = l$m[i]
  }
  # print(logpost(lnew,x,h))
  if (lnew$m[2]<lnew$m[1]){
    ##si les m se sont croisés, on échange tout : m, tau, p et z
    lnew$m = rev(lnew$m)
    lnew$tau = rev(lnew$tau)
    lnew$p = 1-lnew$p
    lnew$z = 3-lnew$z
  }
  # print(logpost(lnew,x,h))
  ##proposition tau
  i = sample(2,1)
  lnew$tau[i] = rgamma(1,inv_prop_tau*l$tau[i]**2,inv_prop_tau *l$tau[i])
  log_ratio_prop = log(dgamma(l$tau[i]     ,inv_prop_tau*lnew$tau[i]**2,inv_prop_tau *lnew$tau[i]))-log(dgamma(lnew$tau[i]     ,inv_prop_tau*l$tau[i]**2,inv_prop_tau *l$tau[i]))
  if (log(runif(1))>(log_ratio_prop+logpost(lnew,x,h)-logpost(l,x,h)) ){
    ##on rejette avec proba 1-alpha. ici on prend en compte le ratio
    lnew$tau[i] = l$tau[i]
  }
  ##proposition z
  # for (i in 1:N){
  # nrunz = 50
  samp = sample(N,nrunz)
  for (i in samp){
    # if (runif(1)<1/2){
      # i = sample(N,1)
      lnew$z[i] = 3-l$z[i]
      if (log(runif(1))>(logpost(lnew,x,h)-logpost(l,x,h)) ){
        ##on rejette avec proba 1-alpha. ici la proposition est symétrique
        lnew$z[i] = l$z[i]
      } else{
        l$z[i] = lnew$z[i]
      }
    # }
  }
  # print(logpost(lnew,x,h))
  ##proposition p
  lnew$p = rbeta(1,inv_prop_p*l$p,inv_prop_p*(1-l$p))
  log_ratio_prop = log(dbeta(l$p,inv_prop_p*lnew$p,inv_prop_p*(1-lnew$p)))-log(dbeta(lnew$p,inv_prop_p*l$p,inv_prop_p*(1-l$p)))
    if (log(runif(1))>(log_ratio_prop+logpost(lnew,x,h)-logpost(l,x,h)) ){
    ##on rejette avec proba 1-alpha. ici on prend en compte le ratio
    lnew$p = l$p
    }
  return(lnew)
}
lnew = step_MH(l,x,h)

```





```{r}
run_MH = function(nMH,x,...){
  linit = init(x)
  l = linit$l
  h = linit$h
  traj = matrix(0,nMH,5)
  colnames(traj) = c("p","m0","m1","tau0","tau1")
  nb = rep(0,length(x))
  for (i in 1:nMH){
    nb = nb+(l$z-1)
    l = step_MH(l,x,h,...)
    traj[i,] = c(l$p,l$m,l$tau)
  }
  class = nb/nMH
  return(list(traj = traj ,class =class,l = l))
}
nMH = 100
ltraj  = run_MH(nMH,x)
plot(1:nMH,ltraj$traj[,2],"l")
lines(1:nMH,ltraj$traj[,3],col = "red")
```

Pour un plot plus joli, on reprend la fonction de la dernière fois
```{r}
plot_run = function(x,mem,l,i = NULL,nbpoints = 10000){
    if (is.null(i)){i = dim(mem)[1]}
    par(mfrow = c(1,2))
    hist(x,50,freq= F,main = "Current estimated density")
    curve((1-l$p)*dnorm(x,l$m1,1/sqrt(l$tau1))+(l$p)*dnorm(x,l$m2,1/sqrt(l$tau2)),add = T,col = "red",lty = 2,lwd=  2)
    curve((1-l$p)*dnorm(x,l$m1,1/sqrt(l$tau1)),add = T,col = "blue",lty = 3,lwd = 2)
    curve((l$p)*dnorm(x,l$m2,1/sqrt(l$tau2)),add = T,col = "green",lty = 3,lwd = 2)
    z1 = sample(i,nbpoints,replace = T)
    z2 = sample(i,nbpoints,replace = T)
    plot(z1,mem[z1,2]+2*1/sqrt(mem[z1,4])*runif(nbpoints,-1,1),ylim = range(x),col = "cyan",cex = 0.1,main = "MH trajectory with 2-sigma bands",ylab = "Trajectories for m1,m2",xlab= "Time")
    points(z2,mem[z2,3]+2*1/sqrt(mem[z2,5])*runif(nbpoints,-1,1),ylim = range(x),col = "yellow",cex = 0.1)
    lines(1:i,mem[1:i,2],ylim = range(x),col = "blue")
    lines(1:i,mem[1:i,3],col = "green")
}

init = function(x){
  h = list(a =1,b = 1, mu = 0,sig = 1000,k = 1,lam = 0.001)
  l = list(p  = 1/2,m = c(0,0),tau = c(1,1),z = 1+rbinom(length(x),1,1/2))
  # l = list(p  = 1/2,m = c(-1,1),tau = c(1,1),z = 1+(x>mean(x)))
  return(list(h = h,l = l))
}

N = 1000
true = list(p = 0.6,m1 = 0,m2 = 2,tau1 =1,tau2 = 5)##pour la simulation
x = rmix(1000,true$p,true$m1,true$m2,true$tau1,true$tau2)
nMH = 2000
sigprop =  c(0.5,0.1,0.01)
nrunz = 50
ltraj  = run_MH(nMH,x,sigprop,nrunz)
l = ltraj$l
lend = list(p = l$p,m1  = l$m[1],m2 = l$m[2],tau1 = l$tau[1],tau2 = l$tau[2])
plot_run(x,ltraj$traj,lend)

```

On peut alors voir les paramètres finaux :

```{r}
cat("estimés : ",unlist(lend),"\n")
cat("théorique :",unlist(true),"\n")
```

On peut aussi observer la propotion du temps que chaque observation a passé avec $Z_i=1$ :

```{r}
s = sort(x,index.return = T)
plot(s$x,ltraj$class[s$ix],"l",main  = "Classification probability",xlab = "x",ylab = "Estimated P(Z(x)= 1)")
```

Et si on veut faire un petit film, il n'y a qu'à modifier les fonctions
```{r}
library(animation)

plot_run = function(x,mem,class,l,i = NULL,nbpoints = 10000,MHtot = NULL){
    if (is.null(i)){i = dim(mem)[1]}
    if (is.null(MHtot)){MHtot = dim(mem)[1]}
    # par(mfrow = c(1,3))
    hist(x,50,freq= F,main = "Current estimated density")
    curve((1-l$p)*dnorm(x,l$m1,1/sqrt(l$tau1))+(l$p)*dnorm(x,l$m2,1/sqrt(l$tau2)),add = T,col = "red",lty = 2,lwd=  3)
    curve((1-l$p)*dnorm(x,l$m1,1/sqrt(l$tau1)),add = T,col = "blue",lty = 3,lwd = 3)
    curve((l$p)*dnorm(x,l$m2,1/sqrt(l$tau2)),add = T,col = "green",lty = 3,lwd = 3)
    z1 = sample(i,nbpoints,replace = T)
    z2 = sample(i,nbpoints,replace = T)
    plot(z1,mem[z1,2]+2*1/sqrt(mem[z1,4])*runif(nbpoints,-1,1),ylim = range(x),xlim = c(1,MHtot),col = "cyan",cex = 0.1,main = "MH trajectory with 2-sigma bands")
    points(z2,mem[z2,3]+2*1/sqrt(mem[z2,5])*runif(nbpoints,-1,1),ylim = range(x),col = "yellow",cex = 0.1)
    lines(1:i,mem[1:i,2],ylim = range(x),col = "blue")
    lines(1:i,mem[1:i,3],col = "green")
    s = sort(x,index.return = T)
    plot(s$x,class[s$ix]/i,"l")
}


run_MH = function(nMH,x,pas_anim,...){
  linit = init(x)
  l = linit$l
  h = linit$h
  traj = matrix(0,nMH,5)
  colnames(traj) = c("p","m0","m1","tau0","tau1")
  nb = rep(0,length(x))
  traj[1,] = c(l$p,l$m,l$tau)
  for (i in 2:nMH){
    nb = nb+(l$z-1)
    l = step_MH(l,x,h,...)
    lnow = list(p = l$p,m1  = l$m[1],m2 = l$m[2],tau1 = l$tau[1],tau2 = l$tau[2])
    if ((i%%pas_anim)==2){
      plot_run(x,traj[1:i,],class = nb,l = lnow,i=i,MHtot =nMH )
      ani.record()
    }
    traj[i,] = c(l$p,l$m,l$tau)
  }
  class = nb/nMH
  return(list(traj = traj ,class =class,l = l))
}
# 
# N = 100
# true = list(p = 0.6,m1 = 0,m2 = 2,tau1 =1,tau2 = 5)##pour la simulation
# x = rmix(1000,true$p,true$m1,true$m2,true$tau1,true$tau2)
# sigprop =  0.1*c(0.5,0.1,0.01)
# nrunz = 20
# nMH = 500
# pas_anim = 10
# par(mfrow = c(1,3))
# ani.options(interval = 0.05, nmax = floor(nMH/pas_anim))
# ani.record(reset = TRUE)
# ltraj  = run_MH(nMH,x,pas_anim= pas_anim,sigprop,nrunz)


# saveGIF(ani.replay(),movie.name = "animation_MH.gif")

```

<img src="animation_MH.gif">



## Commentaires et extensions


On peut aussi observer la chose suivante :

$$\begin{align*}
\pi(m_1|X,Z,\tau,p) &\propto \pi(m_1)\prod_{i = 1}^N \Bigg(\exp \bigg(- \frac{\tau_1(x_i-m_1)^2}{2}\bigg) \Bigg)^{1\!\!1_{z_i = 1}}
\\ & \propto \exp\Bigg(-\frac{(m_1-\mu_0)^2}{2\sigma_0^2}- \sum_{i = 1}^N 1\!\!1_{z_i = 1} \frac{\tau_1(x_i-m_1)^2}{2}\Bigg) 
\end{align*}$$


Par propriété de conjugaison de loi, on voit alors que 
$$m_1|X,Z,\tau,p \sim \mathcal{N}\Bigg(\frac{\frac{\mu_0}{\sigma_0^2} +\tau_1 \sum_{i = 1}^N 1\!\!1_{z_i = 1}x_i }{\frac{1}{\sigma_0^2}+\tau_1 \sum_{i = 1}^N 1\!\!1_{z_i = 1} } ,\frac{1}{ \frac{1}{\sigma_0^2}+\tau_1 \sum_{i = 1}^N 1\!\!1_{z_i = 1}} \Bigg) $$

Et de même pour tous les autres paramètres. Cela incite à reprendre tout notre code pour calculer seulement cela à chaque étape, sans devoir recalculer la vraisemblance.

Voilà un bon exercice pour votre projet !

## Et avec les packages 



```{r}
setwd("/home/espinasse/Bureau/TAF/Seafile/Cours/stat bayesiennes/tp R 2020/old_site")
library(rjags)
N = 1000
true = list(p = 0.6,m1 = 0,m2 = 2,tau1 =1,tau2 = 5)##pour la simulation
u = rmix(10000,true$p,true$m1,true$m2,true$tau1,true$tau2)
m <- jags.model("model_mixture.bug", data=list(x = u, N=N))

f = coda.samples(m,c("m","tau","p"),n.iter = 1000)
traj = f[[1]]
summary(f)
par(mfrow = c(5,2))
# for (i in 1:5){
#   plot(1:nrow(traj),traj[,i],"l",main = colnames(traj)[i])
# }
plot(f)
```


